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Popular Calculus >

taylor 1/(sin(x))

Solution

taylor

\frac{1}{sin ( x )}

Solution

\frac{1}{sin ( 1 )} - cot (1) csc (1) (x - 1) + \frac{- csc ( 1 ) + 2 csc ^{3} ( 1 )}{2} (x - 1)^{2} + \frac{cot ( 1 ) csc ( 1 ) - 6 csc ^{3} ( 1 ) cot ( 1 )}{6} (x - 1)^{3} + \frac{csc ( 1 ) - 2 csc ^{3} ( 1 ) - 6 ( - 3 csc ^{3} ( 1 ) cot ^{2} ( 1 ) - csc ^{5} ( 1 ) )}{24} (x - 1)^{4} + \dots

Solution steps

Apply the Taylor Formula

= \frac{1}{sin ( 1 )} + \frac{\frac{d}{d x} ( \frac{1}{s i n ( x )} ) ( 1 )}{1 !} (x - 1) + \frac{\frac{d ^{2}}{d x ^{2}} ( \frac{1}{s i n ( x )} ) ( 1 )}{2 !} (x - 1)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( \frac{1}{s i n ( x )} ) ( 1 )}{3 !} (x - 1)^{3} + \dots

Evaluate Derivatives

= \frac{1}{sin ( 1 )} + \frac{- cot ( 1 ) csc ( 1 )}{1 !} (x - 1) + \frac{- csc ( 1 ) + 2 csc ^{3} ( 1 )}{2 !} (x - 1)^{2} + \frac{cot ( 1 ) csc ( 1 ) - 6 csc ^{3} ( 1 ) cot ( 1 )}{3 !} (x - 1)^{3} + \frac{csc ( 1 ) - 2 csc ^{3} ( 1 ) - 6 ( - 3 csc ^{3} ( 1 ) cot ^{2} ( 1 ) - csc ^{5} ( 1 ) )}{4 !} (x - 1)^{4} + \dots

Refine

= \frac{1}{sin ( 1 )} - cot (1) csc (1) (x - 1) + \frac{- csc ( 1 ) + 2 csc ^{3} ( 1 )}{2} (x - 1)^{2} + \frac{cot ( 1 ) csc ( 1 ) - 6 csc ^{3} ( 1 ) cot ( 1 )}{6} (x - 1)^{3} + \frac{csc ( 1 ) - 2 csc ^{3} ( 1 ) - 6 ( - 3 csc ^{3} ( 1 ) cot ^{2} ( 1 ) - csc ^{5} ( 1 ) )}{24} (x - 1)^{4} + \dots

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